Ring of Polynomial Forms over Field is Vector Space

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Theorem

Let $\struct {F, +, \times}$ be a field whose unity is $1_F$.

Let $F \sqbrk X$ be the ring of polynomials over $F$.


Then $F \sqbrk X$ is an vector space over $F$.


Corollary

Let $S \subseteq F \sqbrk X$ denote the subset of $F \sqbrk X$ defined as:

$S = \set {\mathbf x \in F \sqbrk X: \map \deg {\mathbf x} < d}$

for some $d \in \Z_{>0}$.

Then $S$ is an vector space over $F$.


Proof

Let the operation $\times': F \to F \sqbrk X$ be defined as follows.

Let $x \in F$.

Let $\mathbf y \in F \sqbrk X$ be defined as:

$\mathbf y = \ds \sum_{k \mathop = 0}^n y_k X^k$

where $n = \map \deg {\mathbf y}$ denotes the degree of $\mathbf y$

Thus:

$x \times' \mathbf y := \ds x \sum_{k \mathop = 0}^n y_k X^k = \sum_{k \mathop = 0}^n \paren {x \times y_k} X^k$

We have that $\times': F \to F \sqbrk X$ is an instance of polynomial multiplication where the multiplier $x$ is a polynomial of degree $0$.


Hence, let the supposed vector space over $F$ in question be denoted in full as:

$\mathbf V = \struct {F \sqbrk X, +', \times'}_F$

where:

$+': F \sqbrk X \to F \sqbrk X$ denotes polynomial addition
$\times': F \to F \sqbrk X$ denotes the operation as defined above.

We already have that $F \sqbrk X$ is an integral domain.

Thus vector space axioms $\text V 0$ to $\text V 4$ are fulfilled.

By definition of $\times'$, it is seen that the remaining vector space axioms are fulfilled as follows:


Let $\lambda, \mu \in F$.

Let $\mathbf x, \mathbf y \in F \sqbrk X$ such that $\map \deg {\mathbf x} = m$ and $\map \deg {\mathbf y} = n$.


Vector Space Axiom $\text V 5$: Distributivity over Scalar Addition

\(\ds \paren {\lambda + \mu} \times' \mathbf x\) \(=\) \(\ds \paren {\lambda + \mu} \times' \sum_{k \mathop = 0}^m x_k X^k\)
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^m \paren {\lambda + \mu} \times x_k X^k\) Definition of $\times'$
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^m \paren {\lambda \times x_k X^k + \mu \times x_k X^k}\) Field Axiom $\text D$: Distributivity of Product over Addition
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^m \lambda \times x_k X^k + \sum_{k \mathop = 0}^m \mu \times x_k X^k\)
\(\ds \) \(=\) \(\ds \lambda \times' \mathbf x + \mu \times' \mathbf x\) Definition of $\times'$

$\Box$


Vector Space Axiom $\text V 6$: Distributivity over Vector Addition

\(\ds \lambda \times' \paren {\mathbf x + \mathbf y}\) \(=\) \(\ds \lambda \times' \paren {\sum_{j \mathop = 0}^m x_j X^j + \sum_{k \mathop = 0}^n y_k X^k}\)
\(\ds \) \(=\) \(\ds \sum_{j \mathop = 0}^m \lambda \times x_j X^j + \sum_{k \mathop = 0}^n \lambda \times y_k X^k\) Definition of $\times'$
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^{\max \set {m, n} } \paren {\lambda \times x_k + \lambda \times y_k} X^k\) Definition of Polynomial Addition
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^{\max \set {m, n} } \lambda \times \paren {x_k + y_k} X^k\) Field Axiom $\text D$: Distributivity of Product over Addition
\(\ds \) \(=\) \(\ds \lambda \times' \mathbf x + \lambda \times' \mathbf y\)

$\Box$


Vector Space Axiom $\text V 7$: Associativity with Scalar Multiplication

\(\ds \lambda \times' \paren {\mu \times' \mathbf x}\) \(=\) \(\ds \lambda \times' \paren {\mu \times' \sum_{k \mathop = 0}^n x_k X^k}\)
\(\ds \) \(=\) \(\ds \lambda \times' \paren {\sum_{k \mathop = 0}^n \mu \times x_k X^k}\) Definition of $\times'$
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^n \lambda \times \paren {\mu \times x_k} X^k\) Definition of $\times'$
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^n \paren {\lambda \times \mu} \times x_k X^k\) Field Axiom $\text M1$: Associativity of Product
\(\ds \) \(=\) \(\ds \paren {\lambda \times \mu} \times' \sum_{k \mathop = 0}^n x_k X^k\) Definition of $\times'$
\(\ds \) \(=\) \(\ds \paren {\lambda \times \mu} \times' \mathbf x\)

$\Box$


Vector Space Axiom $\text V 8$: Identity for Scalar Multiplication

\(\ds 1_F \times' \mathbf x\) \(=\) \(\ds 1_F \times' \sum_{k \mathop = 0}^n x_k X^k\)
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^n 1_F \times x_k X^k\) Definition of $\times'$
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^n x_k X^k\) Field Axiom $\text M3$: Identity for Product
\(\ds \) \(=\) \(\ds \mathbf x\)

$\Box$


All vector space axioms are hence seen to be fulfilled.

Hence the result.

$\blacksquare$


Sources

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